Apple - Is there a way to update the emoji font?

The font file you need to use a more recent version of is /System/Library/Fonts/Apple Color Emoji.ttc and is about 135MB.

The High Sierra version of the file and the process to install it to /Library/Fonts can be found here: http://stephenradford.me/install-high-sierra-emoji-on-older-versions/

Note that the Emoji & Symbol picker will not list the newer emoji added since your version of macOS. For that to work you would also need to update at least /System/Library/PrivateFrameworks/CoreEmoji.framework and maybe some other files.

You have to either obtain a copy of the newer Apple Color Emoji font from a newer OS version and install it in your machine, or manually edit the one you have using a font editor app which can handle this kind of font. I think the second option is not very practical for most users in terms of the work and skills required.